An application of alternative risk measures to trading porfolios

An application of alternative risk measures to trading porfolios

Master Thesis: An Application of Alternative Risk Measures to Trading Portfolios An Application of Alternative Risk Measures to Trading Portfolios Master of Advanced Studies in Finance ETH/Uni Zurich 30 January 2004 Abstract This study covers the advantages of expected shortfall as an alternative risk measure to value-at-risk and the results of implementing in practice the tools of extreme value theory EVT is applied to a varied sample of trading portfolios across different sectors and sensitive to one and multiple risk factors A detailed analysis of the tail of the profit & loss empirical distribution is performed with an emphasis on the estimates of value-at-risk and expected shortfall The concept of expected shortfall is also used as a measure of sensitivity of the portfolio to risk factors, thus allowing to determine the main drivers of risk Being involved with the market directly and on a daily basis, as well as considering the recent events in the Russian market - more specifically, the Yukos case, provided the opportunity to observe a real example when historical VaR fails to be coherent Cornelia Glavan Supervisor: Prof Dr Uwe Schmock Institute for Financial and Actuarial Mathematics Vienna University of Technology Supervisor: Dr Andreas Bitz Head Market Risk Control UBS Investment Bank Switzerland -1- Master Thesis: An Application of Alternative Risk Measures to Trading Portfolios Acknowledgements A master thesis is often perceived as a result of an individual effort This is hardly the case here, as the following study is a result of a team work The paper was written while doing an internship with the Market Risk Control at UBS Investment Bank Zurich I am highly appreciative of the guidance and insights into the business from my supervisor Andreas Bitz in particular, and the help from the whole Market Risk Control team in general Sincere thanks to Roberto Frassanito and Michael Rey for their most useful explanations and remarks I am very grateful to Prof Uwe Schmock for all the help with the mathematical theory And last but not least, many thanks to all my friends and family for their help and support -2- Master Thesis: An Application of Alternative Risk Measures to Trading Portfolios Contents Abstract……………………………………………………………………………… Overview…………………………………………………………………………… Mathematical Theory………………… ………………………………………… 1.1 Expected Shortfall………………… ………………………………………….6 1.2 Generalized Pareto Distribution…….…………………………………… … 1.3 Method of Block Maxima…………………………………………………… 1.4 Historical Approach …… ….…………………………………………….…10 Coherence and VaR………………………………………………………… …11 2.1 Academic Example…………………………………………… …………….11 2.2 Practical Example: … ………………………………………………………11 2.3 Exploring the Tail………………………………………………………….…13 Equity Portfolios………………………………………………………………….15 3.1 One Risk Factor………………………………………………………………15 3.1.1 UBS stocks.…………………………………… ………………….…15 3.1.2 Comparative analysis…………………………………….………… 17 3.2 Two Risk Factors…………………………………………………………….19 Currency and Fixed Income Portfolios: Multiple Risk Factors……………… …21 4.1 Currency portfolios……………………………………….….……………….21 4.2 Fixed Income Portfolios…………………………………………………… 21 4.2.1 One Risk Factor: the Yield………………………………………… 22 4.2.2 Multiple Risk Factor ……………………………………………… 23 Mixed Multiple Risk Factor Portfolio……… ……………………………….….25 Conclusions: Why Expected Shortfall?………………… …………….….…………29 References……………………… ………………………….…………………….….30 Appendix B…………………………………………………………… ……… …31 Appendix C…………………………………………………………….………… 32 Appendix D…………………………………………………………….……… … 34 Appendix E…………………………………………………………… ……… … 36 -3- Master Thesis: An Application of Alternative Risk Measures to Trading Portfolios Overview Following Basel I rules, value-at-risk (VaR) has been established as one of the main risk measures Although widely used by financial institutions, the risk measure is heavily criticized in the academic world for not being sub-additive, i.e the risk of a portfolio as a whole can be larger then the sum of the stand-alone risks of its components when measured by VaR Consequently, VaR may fail to justify diversification and does not take into account the severity of an incurred damage event As a response to these deficiencies, the notion of coherent risk measures was introduced The most well known coherent risk measure is expected shortfall (ES), which is the expected loss provided that the loss exceeds VaR A more detailed theoretical explanation is given in the first chapter, which comprises the mathematical theory that is used in this paper The current method at most financial institutions in estimating VaR is based on a historical framework In order to challenge this method, extreme value theory (EVT) is used to estimate both value-at-risk and expected shortfall The method of EVT focuses on modelling the tail behaviour of a loss distribution using only extreme values rather than all the data This method, generalized Pareto distribution (GPD), has the theoretical background which allows fitting the tail of the losses to a certain class of distributions The second chapter provides two examples demonstrating the advantages of expected shortfall over VaR: the first is an artificial example, while the second is taken from practice and follows exactly the framework currently used in estimating VaR at most financial institutions The last example in this chapter represents a practical case where extreme value theory can be used for financial data using a different method, the one of block Maxima This method is applied in order to give a better understanding of event risk and is able to provide answers to questions like: "How rare is an event of obtaining a return as low or lower than a certain loss?" Further the paper contains the results of applying EVT to equity portfolios, currency portfolios and fixed income portfolios The third chapter discusses the results of applying EVT to equity portfolios consisting of a single position: long stock and short stock In these cases, we have only one risk factor the price log returns To facilitate a better understanding of the behaviour of generalized Pareto distribution when applied to equity portfolios, a representative collection of a wide range of stocks was chosen: SMI and DAX, the well diversified European indices, Nasdaq - the much more volatile than its European counter-parties American index And the stocks of a highly liquid financial institution, and the highly volatile stocks of ABB, Disetronic and Yukos, which have undergone a lot of distress in the time period considered Next the analysis is done on positions of being long, short option on SMI Two risk factors are involved in these cases: the price log returns of the underlying and the absolute returns of the implied volatility EVT is applied in estimating the risk measure for the profit and loss function (P&L) when risk factors are considered individually, as well as for the aggregated P&L Given the multitude of risk factors involved, historical P&L is preferred because it makes no assumptions about the correlation between them The forth chapter discusses the results of estimating VaR and ES for some representative portfolios from the currency and fixed income sectors In an attempt to cover a different gamut of currency portfolios, two low volatile cases (JPY/USD and USD/EURO) and a highly volatile low liquidity case, characterized by an emerging market currency USD/TRL, are considered Generalized Pareto distribution is fitted to both the upper and the lower tail of the distributions The chapter continues with a summary of the results of fitting the tail to GPD for fixed income portfolio; two different bonds are considered The main risk factors to which the P&L is mapped in the first case is the spread to the Treasury curve, and the LIBOR curve and the spread in the second one The idea behind expected shortfall was used to measure the sensitivity of the aggregated P&L to the moves of different risk factors -4- Master Thesis: An Application of Alternative Risk Measures to Trading Portfolios The fifth chapter considers a hypothetical portfolio containing a variety of financial instruments covering all the different businesses discussed in the paper The historical approach is used to compute the P&Ls mapped on different risk factors and expected shortfall is used as the main risk measure, which allows us to make no assumptions about the correlation between risk factors and to make sure that cases of incoherence are being avoided The concept of expected shortfall is successfully used to determine the main drivers of risk in the portfolio This tool is employed to measure the sensitivity of the overall portfolio to individual risk factors, thus allowing us to have a clear view of the risk and potentially point to hedging strategies -5- Master Thesis: An Application of Alternative Risk Measures to Trading Portfolios Chapter Mathematical Theory In this chapter the basic mathematical definitions necessary in understanding the paper are given and also a detailed description of the methods used One can skip this chapter now, provided that while going through the paper you can then come back to check the notions used, given that links are provided to find easier the parts of interest 1.1 Expected Shortfall Expected shortfall was proposed as an alternative risk measure to VaR, having the main property of being coherent In many articles different definitions for ES can be noticed, mainly coming to the same one when we have the assumption that the distribution of the loss is continuous, differences can appear when the distribution of the loss is no longer continuous That is why it is important to have strict definitions of expected shortfall and other risk measures For those interested in exploring further into this topic the article “On the Coherence of Expected Shortfall” by Carlo Acerbi and Dirk Tasche provides all the necessary analytics and proofs used here, but not covered, which is beyond the purpose of this paper Let X be a real-valued random variable (r.v.) on a fixed probability space X is considered the random profit of the portfolio, so we are mainly interested in losses, i.e low values of X The exact mathematical definitions of some risk measures are given bellow: α - the confidence level (usually 0.95 or 0.99) Definition 1: Value-at-Risk: gives the maximum loss such that with a (1-α) probability we exceed it VaRα ( X ) = − inf{x ∈ R : P[ X ≤ x] ≥ − α } (1.1.1) Observation: this is actually the lower (1-α)-quantile of X, taken with minus Definition 2: Tail Conditional Loss: The expected loss provided that the loss exceeds VaR TCLα ( X ) = − E ( X X ≤ −VaRα ( X ) ) (1.1.2) In some papers this is used as the definition for expected shortfall But we need to be careful about it, because this is the case when the distribution of the loss is continuous The main property of Expected shortfall is coherence, but using the definition as in the tail conditional loss (TCL) case, the property of coherence is lost, as it is shown in the first example from the next chapter Before proceeding with the definition of expected shortfall, it is good to give the exact characteristics of the property of coherence, especially that intuitively this property is expected to be fulfilled by any function which gives us a number as a measure of risk Before proceeding with the concept of coherence we give the strict definition for a risk measure: Definition 3: Risk Measure Let V be a set of real random variables on some probability space such that E[X-]1< ∞ for all X∈V Then a function ρ, which a gives a real number for any random variable in V is called a risk measure ρ: V → R It is natural from a practical point of view to define coherence in the following way: X − = − X 1{ X 0, hX∈V then ρ(hX) = hρ(X) The risk is increasing proportionally with the magnitude of the portfolio, given the weights of the assets stay the same (iv) translation invariant: X∈V, a – real number then ρ(X + a) = ρ(X) - a If a certain gain (loss) is added to the portfolio, then its risk decreases (increases) with the same amount as the gain (loss) Definition 4: Expected Shortfall: ES α ( X ) = − ( E[ X 1{ X ≤ −VaRα ( x )} ] − VaRα ( X )((1 − α ) − P[ X ≤ −VaRα ( X )])) (1.1.3) 1−α As can be seen, by comparing ES with TCL, they are equal when the distribution of the loss is continuous, so we may say that for continuous distributions TCL is coherent, problems arise when dealing with cases in which P [X ≤ -VaRα (X)] ≠ 1-α Expected Shortfall is always coherent An example showing us that VaR and TCL are not coherent is provided in the second chapter VaR on the other hand satisfies most of the properties of being a coherent risk measure, except for the case of the sub-additive property VaR is sub-additive for example in the case when the distribution of X is elliptical, as it is the case of the normal and t-student distributions For more details on the subject about the coherence of VaR for elliptical distributions the reader is referred to [8] 1.2 Generalized Pareto Distribution Extreme value theory goes back to the late 1920 But only recently it gained recognition as a practical and useful tool in estimating tail risk In the middle of the last century the econometricians already discovered the non-normal behavior of financial markets, but this assumption is still widely used in the financial industry, provided it is easy to implement Using the EVT method we look only at extreme losses and under the assumption that the losses occur independently we are also given the theoretical background to use it Here is discussed the generalized Pareto distribution used in estimating the tail, this method is also referred to in many articles as peaks over threshold A small theory is given and also the Balkema and De Haan theorem on which the extreme value theory is constructed A threshold u is chosen, and then the losses, which lie beyond it are fitted to a GPD The tools of better choosing the threshold and the method to estimate the GPD used in this paper are discussed further Let X1 , X2 , X3 , …, Xn be the losses We make the assumption of them being independent and identically distributed, and let F(x) = P (X1 ≤ x) be their distribution function Definition 5: Let xF be the right end point of the distribution F x F = sup{x ∈ R F ( x) < 1} ≤ ∞ -7- (1.2.1) Master Thesis: An Application of Alternative Risk Measures to Trading Portfolios Definition 6: For any u < xF, we define the distribution function of the excesses over the threshold u by Fu ( x) = P( X − u ≤ x X > u ) = F ( x + u ) − F ( x) − F (u ) (1.2.2) Comment: the choice of the threshold u being smaller then the right end point of the distribution of the loss insures the fact that the probability of having a loss that exceeds it, is positive We can now discuss the maximum domain of attraction (MDA) conditions MDA conditions: Using all of the assumptions before, let Mn = max {X1, X2,…, Xn } Suppose that there exist the sequences of strictly positive numbers (a n ) n∈N and a sequence of real numbers (bn ) n∈N such that the sequence of transformed maxima ( M n − bn ) n converges in an distribution  M − bn  →∞ P n ≤ x  = F n (a n x + bn ) n → H ξ ( x) ,for every continuity point x of Hξ, (1.2.3)  an  where Hξ is a non-degenerate distribution function Comm ent: Reflecting on the MDA conditions the following question arises: “How does the normalizing sequences (a n ) n∈N and (bn ) n∈N influence the limiting distribution Hξ, ?” The answer is that the limit law is uniquely determined up to affine transformations The proof is provided in the book “Modelling Extremal Events for Insurance and Finance” by P Embrechts, C Klüppelberg and T Mikosch, theorem A1.5 Now we are ready to write the fundamental theorem Theorem Balkema and De Haan (1974) Under the MDA conditions, the generalized Pareto distribution is the limiting distribution of the excesses, as the threshold tends from below to the right-end point That is, there exists β (u) – positive function of u such that: lim sup Fu ( x) − Gξ , β (u ) ( x) = u < x F , u → x F ≤ x ≤ x −u F (1.2.4) with Gξ,β(u) defined:  x −ξ ) ,ξ ≠ 1 − (1 + ξ β (u ) Gξ , β ( u ) ( x) =  x −  β (u )  1− e ,ξ = where: x ≥ when ξ ≥ 0, and ≤ x ≤ -β (u)/ ξ when ξ < (1.2.5) We say that F belongs to the maximum domain of attraction of Hξ, where Hξ is the generalized extreme value distribution (GEVD): −  exp{−(1 + ξx) ξ }, ξ ≠ H ξ ( x) =   exp{−e − x }, ξ = (1.2.6) where: 1+ξx > In fact this is the limiting distribution for transformed maxima from (1.2.3) The parameter ξ is called the shape parameter, and it is an indicator of the fatness of the tail -8- Master Thesis: An Application of Alternative Risk Measures to Trading Portfolios 1.2.1 Properties of the tail of the distribution F: If ξ>0, then we say that F belong to the MDA of the Fréchet distribution Gnedenko showed (1943) that the tail decays like a power function In this class are heavy tail distributions like Pareto, log-Gamma, Cauchy and t-student If ξ=0, then F is the MDA of the Gumbel distribution, this is the case of distributions where the tail decays like the exponential function, examples: normal, exponential, and lognormal If ξ u the upper tail distribution is the following: F ( x) = (1 − P( X ≤ u )) Fu ( x − u ) + P( X ≤ u ) (1.2.7) from here, by taking Fn(u) as the empirical probability of not exceeding the threshold (number of losses smaller the u over the total number of losses) we can estimate the upper tail distribution:  F ( x) = (1 − Fn (u ))Gξ , β ( x) + Fn (u ) (1.2.8) It can be shown that F(x), for x > u can also be approximated with a GPD with the same shape parameter ξ as for Gξ,β(u) , the fitted GPD to the distribution of the excesses Method used to fit the tail: We have a sample of iid (independent and identically distributed) losses x1 , x2 , x3 , …, xn , then we choose a threshold u and we look at extreme losses, those for which x > u Let’s assume there are k losses bigger then u, and we write them as: x’1 , x’2 , …, x’k Define yi = x’i – u, for every i from to k Now we think of y1 , y2 ,…, y3 as being a sample from a GPD with parameters ξ, β which we want to estimate We use the log-likelihood function in estimating the parameters Under the assumption that ξ ≠ 0, the log-likelihood function is: ξy k l (ξ , β ) = −k ln( β ) − (1 + )[∑ ln(1 + i )] ξ i =1 β and so in order to find ξ and β we maximize the log-likelihood function with the restrictions: ξ > and β > or ξ < and β > − max{ y1 , , y k }ξ Let ξ’ and β ’ be the estimates obtained, then the estimate for the tail distribution  x ≥ u , when   ξ >0  k ξ ( x − u ) − ξ  β (1.2.8) F ( x) = − (1 +  ) , for  x ∈ [u , u −  ), whenξ < n β  ξ From here, since VaRα (α-the confidence interval) is the α-quantile, so VaRα =F-1(α) As to avoid any confusions regarding VaR and Definition 1, here we say VaR is a quantile, because we estimate it from the GPD, which is a continuous distribution We say VaRα is the α-quantile and not (1-α)-quantile because in this chapter we fit the tail distribution of losses, while in the first subchapter we considered X - the profit The obtained estimate for VaRα is:   n −ξ k β VaRα = u + [((1 − α ) ) − 1]  , providedα ≥ − k n ξ (1.2.9) And in the case of expected shortfall, since we look at the losses and the distribution is continuous: ESα ( X ) = E[ X X > VaRα ( X )] = VaRα ( X ) + E[ X − VaRα ( X ) X > VaRα ( X )] we obtain the estimate:   ˆaR V β − βu α Eˆ S α =  +  , for the case ξ > 1− ξ 1−ξ -9- (1.2.10) Master Thesis: An Application of Alternative Risk Measures to Trading Portfolios 1.2.3 Mean Excess Function As can been seen from the description above, we need to choose a threshold What is the best method in choosing it? On one hand the more points we have above the threshold, the more points we have in estimating the tail, but on the other hand the theorem tells us the threshold should tend to the right-end point In this case a good indicator in choosing the threshold is the mean excess plot Suppose the threshold excess X-u follows a GPD with parameters ξ and β , under necessary restrictions as in 1.2.5, then E[ X − u X > u ] = ∫ xdGξ , β ( x) = β 1−ξ (1.2.11) For any u’>u we define the mean-excess plot as: e(u ' ) = E[ X − u ' X > u ' ] = β (u ) + ξ (u '−u ) β (u ) ξ = + y 1−ξ 1−ξ 1−ξ (1.2.12) from the equality (1.2.7), it is seen that the mean-excess plot is a linear function of y = u’-u If we take the empirical mean-excess plot:  en (u ' ) = nu ' nu ' ∑ (x i =1 (i ) − u ' ) , then, after plotting it for each u’, we choose u so that for u’ > u it should look like a line In most of the cases studied the conclusion was to choose u’ so that 95% of the sample points lie above u’ 1.3 Method of Block Maxima In this paper a wide range of stocks are used and there will be cases in which the tail estimates obtained seem to be sensible to some very extreme losses And in this case a natural question arises: is this loss an event risk? How rare is this loss and what is the probability of this happening again? EVT can be successfully used to answer this question And it even allows us to loosen the assumption of the losses being independent The general idea is to group the losses in blocks, like by month or quarter And then we fit the maximum losses from each block to the generalized extreme value distribution (1.2.6) This is due to the Fisher-Tippett theorem (which tells us that if F (the distribution of losses) belongs to the maximum domain of attraction then the block maxima follow a generalized extreme value distribution.) The example when this method is used is provided in the second chapter, while here we continue with a small description of the theory applied in it One of the important assumptions allowing for a fitting of block maxima2 to a generalized extreme value distribution (see 1.2.6) is the fact that the losses should be independent and identically distributed Further, the effects of relaxing the assumption of iid to consider just stationary processes3 is considered With additional assumptions it can be shown that normalized block maxima indeed follow a GEV distribution asymptotically We assume (Xn) to be our stationary process, F the marginal distribution of Xi, while (Yn) is the associated iid process with the same marginal distribution F The conditions necessary to be fulfilled by (Xn) such that the maxima of (Xn) have exactly the same limiting behaviour as maxima of (Yn ) are as follows: i) If the stationary series (Xn) shows only weak long-range dependence, so that we can assume that block maxima are independent ii) If it shows no inclination to form clusters of large values Then maxima of the two series have identical limiting behaviour in our cases monthly minimums will be considered as block maxima a stationary process is one which is time invariant: for any h1 < h2 < … hn, and t > we have that: (Xh1, Xh2, …, Xhn) = (in distribution)= (Xh1+t , Xh2+t , …, Xhn+t ) -10- ... transformed maxima from (1.2.3) The parameter ξ is called the shape parameter, and it is an indicator of the fatness of the tail -8- Master Thesis: An Application of Alternative Risk Measures to. .. thus allowing us to have a clear view of the risk and potentially point to hedging strategies -5- Master Thesis: An Application of Alternative Risk Measures to Trading Portfolios ... for standard deviation used was the unbiased estimator and it is equal to 0.018 For normal distribution this ratio is 1.146 -18- Master Thesis: An Application of Alternative Risk Measures to Trading